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Statistical properties of nuclei: beyond the mean field
Yoram Alhassid (Yale University)
• Introduction
• Beyond the mean field: correlations via fluctuations
The static path approximation (SPA)
The shell model Monte Carlo (SMMC) approach.
• Partition functions and level densities in SMMC.
• Level densities in medium-mass nuclei: theory versus experiment.
• Projection on good quantum numbers: spin, parity,…
• A theoretical challenge: the heavy deformed nuclei.
• Conclusion and prospects.
Introduction
Statistical properties at finite temperature or excitation energy:
level density and partition function, heat capacity, moment of inertia,…
• Level densities are an integral part of the Hauser-Feshbach theory
of nuclear reaction rates (e.g., nucleosynthesis)
• Partition functions are required in the modeling of supernovae and stellar
collapse
• Study the signatures of phase transitions in finite systems.
A suitable model is the interacting shell model: it includes both shell effects
and residual interactions.
However, in medium-mass and heavy nuclei the required model space is
prohibitively large for conventional diagonalization.
Beyond the mean field: correlations via fluctuations
Non-perturbative methods are necessary because of the strong interactions.
Mean-field approximations are tractable but often insufficient.
Correlation effects can be reproduced by fluctuations around the mean field
H /T
Gibbs ensemble e
at temperature T can be written as a superposition
of ensembles U of non-interacting nucleons in time-dependent fields
e H D G U
(Hubbard-Stratonovich transformation).
• Static path approximation (SPA): integrate over static fluctuations of the
relevant order parameters.
• Shell model Monte Carlo (SMMC): integrate over all fluctuations by
Monte Carlo methods.
Lang, Johnson, Koonin, Ormand, PRC 48, 1518 (1993);
Alhassid, Dean, Koonin, Lang, Ormand, PRL 72, 613 (1994).
Enables calculations in model spaces that are many orders of magnitude larger.
Level densities
Experimental methods: (i) Low energies: counting; (ii) intermediate
energies: charged particles, Oslo method, neutron evaporation; (iii)
neutron threshold: neutron resonances; (iv) higher energies: Ericson
fluctuations.
Good fits to the data are obtained using the
backshifted Bethe formula (BBF):
Ex
12
a 1/ 4 Ex
5/ 4
e
2 a Ex
a = single-particle level density
parameter.
= backshift parameter.
But: a and are adjusted for each nucleus
and it is difficult to predict
SMMC is an especially suitable method for microscopic calculations: correlation
effects are included exactly in very large model spaces (~1029 for rare-earths)
Partition function and level density in SMMC
[H. Nakada and Y.Alhassid, PRL 79, 2939 (1997)]
Partition function: calculate the thermal energy E ( ) H and
integrate ln Z / E ( ) to find the partition function Z ( )
Level density: the average level density is found from Z ( ) in the saddlepoint approximation:
E
S(E) = canonical entropy;
S ( E ) ln Z E
1
2 T C
2
eS E
C = canonical heat capacity.
C 2E /
Medium mass nuclei (A ~ 50 -70)
[Y.Alhassid, S. Liu, and H. Nakada, PRL 83, 4265 (1999)]
• Complete fpg9/2-shell, pairing plus surface-peaked multipole-multipole
interactions up to hexadecupole (dominant collective components).
SMMC level densities are well fitted
to the backshifted Bethe formula
Extract
•
a
a
and
is a smooth function of A.
• Odd-even staggering effects
in (a pairing effect).
• Good agreement with experimental data without adjustable parameters.
• Improvement over empirical formulas.
Dependence on good quantum numbers
(i) Spin projection
[Y. Alhassid, S. Liu and H. Nakada, Phys. Rev. Lett. 99, 162504 (2007) ]
Spin distributions in even-odd, even-even and odd-odd nuclei
Spin cutoff model:
J
2 J 1 J ( J 1) / 2
e
3
2 2
2
• Spin cutoff model works very well
except at low excitation energies.
• Staggering effect in spin for
even-even nuclei.
Moment of inertia
Thermal moment of inertia can be extracted from:
2
IT
2
Signatures of pairing correlations:
• Suppression of moment of inertia at low excitations in even-even nuclei.
• Correlated with pairing energy of J=0 neutrons pairs.
A simple model
[Y. Alhassid, G.F. Bertsch, L. Fang, and S. Liu; Phys. Rev. C 72, 064326 (2005)]
Model: deformed Woods-Saxon potential plus pairing interaction.
(i) Number-parity projection : the major odd-even effects are described
by a number-parity projection
1
ˆ
Ph 1 h ei N
2
•
Projects on even (h = 1) or odd (h = -1) number of particles.
is obtained from
fk fk 1 e
Ek / T
1
by the replacement
(negative occupations !)
(ii) Static path approximation (SPA)
• include static fluctuations of the pairing order parameter.
iron isotopes (even-even and even-odd nuclei)
• Good agreement with SMMC
• Strong odd/even effect
(ii) Parity projection
H. Nakada and Y.Alhassid, PRL 79, 2939 (1997);
PLB 436, 231 (1998).
even at neutron resonance
energy (contrary to a common assumption
used in nucleosynthesis).
A simple model (I)
Y. Alhassid, G.F. Bertsch, S. Liu, H. Nakada
[Phys. Rev. Lett. 84, 4313 (2000)]
• The quasi-particles occupy levels with parity
according to a Poisson distribution.
Z / Z tanh f
f is the mean occupation of
quasi-particle orbitals with parity
Ratio of odd-to-even parity level
densities versus excitation energy.
A simple model (II)
[H. Chen and Y. Alhassid]
• Deformed Woods-Saxon potential plus pairing interaction.
• Number-parity projection, SPA plus parity projection.
Ratio of odd-to-even parity
partition functions
Ratio of odd-to-even parity
level densities
Extending the theory to higher temperatures/excitation energies
[Alhassid, Bertsch and Fang, PRC 68, 044322 (2003)]
• It is time consuming to include higher shells in the fully correlated calculations.
We have combined the fully correlated partition in the truncated space with the
independent-particle partition in the full space (all bound states plus continuum)
• BBF works well up to T ~ 4 MeV
Extended heat capacity (up to T ~ 4 MeV)
Experiment (Oslo)
Theory (SMMC)
• Strong odd/even effect: a signature of pairing phase transition
A theoretical challenge: the heavy deformed nuclei
[Y. Alhassid, L. Fang and H. Nakada, arXive:0710.1656 (PRL, 2008)]
• Medium-mass nuclei:
small deformation, first excitation ~ 1- 2 MeV in even-even nuclei.
• Heavy nuclei:
large deformation (open shell), first excitation ~ 100 keV, rotational bands.
• Conceptual difficulty:
Can we describe rotational behavior in a truncated spherical shell model?
• Technical difficulties:
Several obstacles in extending SMMC to heavy nuclei.
Example: 162Dy (even-even)
• Model space includes 1029 configurations ! (largest SMMC calculation)
• J 2 versus T confirms rotational
character with a moment of inertia:
J 2 2IT
with I 35.5 3.3MeV 1
(experimental value is 37.2MeV 1 ).
• SMMC level density is in excellent
agreement with experiments.
Rotational character can be reproduced
in a truncated spherical shell model !
Even-odd and odd-odd rare-earth nuclei
(Ozen, Alhassid, Nakada)
• A sign problem when projecting on odd number of particles (at low T)
Conclusion
• Fully microscopic calculations of statistical properties of nuclei are now
possible by the shell model quantum Monte Carlo methods.
• The dependence on good quantum numbers (spin, parity,…) can be
determined using exact projection methods.
• Simple models can explain certain features of the SMMC spin and parity
distributions.
• SMMC successfully extended to heavy deformed nuclei: rotational
character can be reproduced in a truncated spherical shell model.
Prospects
• Systematic studies of the statistical properties of heavy nuclei.
• Develop “global” methods to derive effective shell model interactions.
DFT -> configuration-interaction shell model map
Correlation energies in N=Z sd shell nuclei
Medium mass nuclei (A ~ 50 -70)
We have used SMMC to calculate the statistical properties of nuclei in the
iron region in the complete fpg9/2-shell.
• Single-particle energies from Woods-Saxon potential plus spin-orbit.
• The interaction includes the dominant components of realistic
effective interactions: pairing + multipole-multipole interactions
(quadrupole, octupole, and hexadecupole).
Pairing interaction is determined to reproduce the experimental gap
(from odd-even mass differences).
Multipole-multipole interaction is determined self-consistently and
renormalized.
• Interaction has a good Monte Carlo sign.
Parity distribution
Alhassid, Bertsch, Liu and Nakada, Phys. Rev. Lett. 84, 4313 (2000)
The distribution to find n particles in single-particle states with
parity is a Poisson distribution:
f
f n f
P ( n)
e
n!
For an even-even nucleus:
Z ( ) n _ odd P(n)
tanh f
Z ( ) n _ even P(n)
Where f a
1
1 e
( Ea )
is the total Fermi- Dirac
occupation in all states with parity
Occupation distribution of the even-parity
orbits ( g9/ 2 ) in 60 Ni
• Deviations from Poisson
distribution for T < 1 MeV
(pairing effect)
The model should be applied for
the quasi-particles:
f
a
1
fa
Ea
1
e
a
Thermal signatures of pairing correlations: summary
Nanoparticles (/d 1 versus nuclei
Heat capacity
Spin susceptibility
Experiment (Oslo)
Moment of inertia
• Pairing correlations (for /d ~1) manifest through strong odd/even effects.
Extending the theory to higher temperatures
[Y.Alhassid., G.F. Bertsch, and L. Fang, Phys. Rev. C 68, 044322 (2003)]
It is time consuming to include higher shells in the Monte Carlo
approach.
We have combined the fully correlated partition in the truncated
space with the independent-particle partition in the full space (all
bound states plus continuum):
(i) Independent-particle model
• Include both bound states and continuum:
• Truncation to one major
shell is problematic for T > 1.5
MeV.
• The continuum is important
for a nucleus with a small
neutron separation energy
(66Cr).
Thermal energy vs. inverse temperature
• Ground-state energy in SMMC
has additional ~ 3 MeV of correlation
energy as compared with
Hartree-Fock-Boguliubov (HFB).
• Results from several experiments
are fitted to a composite formula:
constant temperature below EM
and BBF above.
• SMMC level density is in excellent
agreement with experiments.
Experimental state density
• An almost complete set of levels
(with spin) is known up to ~ 2 MeV.
(i) A constant temperature formula is
fitted to level counting.
(ii) A BBF above EM is determined by
matching conditions at EM
A composite formula
(iii) Renormalize Oslo data by fitting
their data and neutron
resonance to the composite formula
The composite formula is an excellent
fit to all three experimental data.